This algorithm [10,8] solves the single-source
shortest-paths problem on a weighted, directed or undirected graph for
the case where all edge weights are nonnegative. Use the Bellman-Ford
algorithm for the case when some edge weights are negative. Use
breadth-first search instead of Dijkstra's algorithm when all edge
weights are equal to one. For the definition of the shortest-path
problem see Section Shortest-Paths
Algorithms for some background to the shortest-path problem.

There are two main options for obtaining output from the
dijkstra_shortest_paths() function. If you provide a
distance property map through the distance_map() parameter
then the shortest distance from the source vertex to every other
vertex in the graph will be recorded in the distance map. Also you can
record the shortest paths tree in a predecessor map: for each vertex
u in V, p[u] will be the predecessor of u in
the shortest paths tree (unless p[u] = u, in which case u is
either the source or a vertex unreachable from the source). In
addition to these two options, the user can provide their own
custom-made visitor that takes actions during any of the
algorithm's event points.

Dijkstra's algorithm finds all the shortest paths from the source
vertex to every other vertex by iteratively ``growing'' the set of
vertices S to which it knows the shortest path. At each step of
the algorithm, the next vertex added to S is determined by a
priority queue. The queue contains the vertices in V - S[1] prioritized by their distance label, which is the
length of the shortest path seen so far for each vertex. The vertex
u at the top of the priority queue is then added to S,
and each of its out-edges is relaxed: if the distance to u plus
the weight of the out-edge (u,v) is less than the distance
label for v then the estimated distance for vertex v is
reduced. The algorithm then loops back, processing the next vertex at
the top of the priority queue. The algorithm finishes when the
priority queue is empty.

The algorithm uses color markers (white, gray, and black) to keep
track of which set each vertex is in. Vertices colored black are in
S. Vertices colored white or gray are in V-S. White vertices have
not yet been discovered and gray vertices are in the priority queue.
By default, the algorithm will allocate an array to store a color
marker for each vertex in the graph. You can provide your own storage
and access for colors with the color_map() parameter.

The following is the pseudo-code for Dijkstra's single-source shortest
paths algorithm. w is the edge weight, d is the distance label,
and p is the predecessor of each vertex which is used to encode
the shortest paths tree. Q is a priority queue that supports the
DECREASE-KEY operation. The visitor event points for the algorithm are
indicated by the labels on the right.

Where Defined

Parameters

The graph object on which the algorithm will be applied.
The type Graph must be a model of
Vertex List Graph
and Incidence Graph.Python: The parameter is named graph.

IN: vertex_descriptor s

The source vertex. All distance will be calculated from this vertex,
and the shortest paths tree will be rooted at this vertex.Python: The parameter is named root_vertex.

Named Parameters

IN: weight_map(WeightMap w_map)

The weight or ``length'' of each edge in the graph. The weights
must all be non-negative, and the algorithm will throw a
negative_edge
exception is one of the edges is negative.
The type WeightMap must be a model of
Readable Property Map. The edge descriptor type of
the graph needs to be usable as the key type for the weight
map. The value type for this map must be
the same as the value type of the distance map.Default:get(edge_weight, g)Python: Must be an edge_double_map for the graph.Python default: graph.get_edge_double_map("weight")

IN: vertex_index_map(VertexIndexMap i_map)

This maps each vertex to an integer in the range [0,
num_vertices(g)). This is necessary for efficient updates of the
heap data structure [61] when an edge is relaxed.
The type
VertexIndexMap must be a model of
Readable Property Map. The value type of the map must be an
integer type. The vertex descriptor type of the graph needs to be
usable as the key type of the map.Default:get(vertex_index, g).
Note: if you use this default, make sure your graph has
an internal vertex_index property. For example,
adjacenty_list with VertexList=listS does
not have an internal vertex_index property.
Python: Unsupported parameter.

OUT: predecessor_map(PredecessorMap p_map)

The predecessor map records the edges in the minimum spanning
tree. Upon completion of the algorithm, the edges (p[u],u)
for all u in V are in the minimum spanning tree. If p[u] =
u then u is either the source vertex or a vertex that is
not reachable from the source. The PredecessorMap type
must be a Read/Write
Property Map whose key and value types are the same as the vertex
descriptor type of the graph.Default:dummy_property_mapPython: Must be a vertex_vertex_map for the graph.

UTIL/OUT: distance_map(DistanceMap d_map)

The shortest path weight from the source vertex s to each
vertex in the graph g is recorded in this property map. The
shortest path weight is the sum of the edge weights along the
shortest path. The type DistanceMap must be a model of Read/Write
Property Map. The vertex descriptor type of the graph needs to
be usable as the key type of the distance map.
The value type of the distance map is the element type of a Monoid formed with the combine
function object and the zero object for the identity
element. Also the distance value type must have a
StrictWeakOrdering provided by the compare function
object.Default:iterator_property_map created from a
std::vector of the WeightMap's value type of size
num_vertices(g) and using the i_map for the index
map.Python: Must be a vertex_double_map for the graph.

IN: distance_compare(CompareFunction cmp)

This function is use to compare distances to determine which vertex
is closer to the source vertex. The CompareFunction type
must be a model of Binary
Predicate and have argument types that match the value type of
the DistanceMap property map.Default:std::less<D> with D=typename
property_traits<DistanceMap>::value_typePython: Unsupported parameter.

IN: distance_combine(CombineFunction cmb)

This function is used to combine distances to compute the distance
of a path. The CombineFunction type must be a model of Binary
Function. The first argument type of the binary function must
match the value type of the DistanceMap property map and
the second argument type must match the value type of the
WeightMap property map. The result type must be the same
type as the distance value type.Default:closed_plus<D> with
D=typename property_traits<DistanceMap>::value_typePython: Unsupported parameter.

IN: distance_inf(D inf)

The inf object must be the greatest value of any D object.
That is, compare(d, inf) == true for any d != inf.
The type D is the value type of the DistanceMap.Default:std::numeric_limits<D>::max()Python: Unsupported parameter.

IN: distance_zero(D zero)

The zero value must be the identity element for the
Monoid formed by the distance values
and the combine function object.
The type D is the value type of the DistanceMap.Default:D()with
D=typename property_traits<DistanceMap>::value_typePython: Unsupported parameter.

UTIL/OUT: color_map(ColorMap c_map)

This is used during the execution of the algorithm to mark the
vertices. The vertices start out white and become gray when they are
inserted in the queue. They then turn black when they are removed
from the queue. At the end of the algorithm, vertices reachable from
the source vertex will have been colored black. All other vertices
will still be white. The type ColorMap must be a model of
Read/Write
Property Map. A vertex descriptor must be usable as the key type
of the map, and the value type of the map must be a model of
Color Value.Default: an iterator_property_map created from a std::vector
of default_color_type of size num_vertices(g) and
using the i_map for the index map.Python: The color map must be a vertex_color_map for
the graph.

OUT: visitor(DijkstraVisitor v)

Use this to specify actions that you would like to happen
during certain event points within the algorithm.
The type DijkstraVisitor must be a model of the
Dijkstra Visitor concept.
The visitor object is passed by value [2].Default:dijkstra_visitor<null_visitor>Python: The parameter should be an object that derives from
the DijkstraVisitor type
of the graph.

Complexity

The time complexity is O(V log V).

Visitor Event Points

vis.initialize_vertex(u, g)
is invoked on each vertex in the graph before the start of the
algorithm.

vis.examine_vertex(u, g)
is invoked on a vertex as it is removed from the priority queue
and added to set S. At this point we know that (p[u],u)
is a shortest-paths tree edge so
d[u] = delta(s,u) = d[p[u]] + w(p[u],u). Also, the distances
of the examined vertices is monotonically increasing
d[u1] <= d[u2] <= d[un].

vis.examine_edge(e, g)
is invoked on each out-edge of a vertex immediately after it has
been added to set S.

vis.edge_relaxed(e, g)
is invoked on edge (u,v) if d[u] + w(u,v) < d[v].
The edge (u,v) that participated in the last
relaxation for vertex v is an edge in the shortest paths tree.

vis.discover_vertex(v, g)
is invoked on vertex v when the edge
(u,v) is examined and v is WHITE. Since
a vertex is colored GRAY when it is discovered,
each reacable vertex is discovered exactly once. This
is also when the vertex is inserted into the priority queue.

vis.edge_not_relaxed(e, g)
is invoked if the edge is not relaxed (see above).

vis.finish_vertex(u, g)
is invoked on a vertex after all of its out edges have
been examined.

Example

See also

Notes

[1]
The algorithm used here saves a little space by not putting all V -
S vertices in the priority queue at once, but instead only those
vertices in V - S that are discovered and therefore have a
distance less than infinity.

[2]
Since the visitor parameter is passed by value, if your visitor
contains state then any changes to the state during the algorithm
will be made to a copy of the visitor object, not the visitor object
passed in. Therefore you may want the visitor to hold this state by
pointer or reference.